MathDB

3

Part of 2011 Korea National Olympiad

Problems(2)

Korea Second Round 2011

Source: Korea Second Round 2011 #3

8/21/2011
Let a,b,c,da,b,c,d real numbers such that a+b+c+d=19a+b+c+d=19 and a2+b2+c2+d2=91a^2+b^2+c^2+d^2=91. Find the maximum value of 1a+1b+1c+1d \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d}
algebrapolynomialfunctioninequalities unsolvedinequalities
Korea Second Round 2011

Source: Korea Second Round 2011 #7

8/21/2011
There are nn students each having rr positive integers. Their nrnr positive integers are all different. Prove that we can divide the students into kk classes satisfying the following conditions.
(a) k≤4r k \le 4r
(b) If a student AA has the number mm, then the student BB in the same class can't have a number ll such that (m−1)!<l<(m+1)!+1 (m-1)! < l < (m+1)!+1
inductioncombinatorics proposedcombinatorics